Borm, Peter

Dietzenbacher, Bas

Musegaas, Marieke

English

Tilburg University Repository

    Tilburg UniversityA Note on Shapley Ratings in Brain NetworksMusegaas, Marieke; Dietzenbacher, Bas; Borm, PeterPublication date:2016Document VersionEarly version, also known as pre-printLink to publication in Tilburg University Research PortalCitation for published version (APA):Musegaas, M., Dietzenbacher, B., & Borm, P. (2016). A Note on Shapley Ratings in Brain Networks. (CentERDiscussion Paper; Vol. 2016-025). CentER, Center for Economic Research.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 01. Nov. 2022                  No. 2016-025    A NOTE ON SHAPLEY RATINGS IN BRAIN NETWORKS     By     M. Musegaas, B.J. Dietzenbacher, P.E.M. Borm      11 July 2016      ISSN 0924-7815 ISSN 2213-9532 A note on Shapley ratings in brain networksM. Musegaas∗† B.J. Dietzenbacher∗ P.E.M. Borm∗July 7, 2016AbstractWe consider the problem of computing the influence of a neuronal structure in abrain network. Abraham, Kötter, Krumnack, and Wanke (2006) computed this in-fluence by using the Shapley value of a coalitional game corresponding to a directednetwork as a rating. Kötter, Reid, Krumnack, Wanke, and Sporns (2007) appliedthis rating to large-scale brain networks, in particular to the macaque visual cortexand the macaque prefrontal cortex. We introduce an alternative coalitional gamethat is more intuitive from a game theoretical point of view. We use the Shapleyvalue of this game as an alternative rating to analyze the macaque brain networksand corroborate the findings of Kötter et al. (2007). Moreover, we show how miss-ing information on the existence of certain connections can readily be incorporatedinto this game and the corresponding Shapley rating.Keywords: brain networks, coalitional games, Shapley value1 IntroductionIn this paper we study the influence of a single neuronal structure on the connectivitystructure of the whole brain network. The aim is to contribute to the methodologyproposed by Abraham et al. (2006) from a game theoretical perspective. Cooperativegame theory analyzes the importance of players in a joint collaboration structure by takinginto account the possibility of cooperation in subgroups or coalitions. Von Neumann andMorgenstern (1944) introduced the model of a coalitional game, in which each coalitionis assigned a worth reflecting what this coalition can achieve if it acts on its own. Inthe context of brain networks, the Shapley value (cf. Shapley (1953)) can be appliedto measure the influence of each neuronal structure in a brain network. This measuredepends on the corresponding coalitional game.In the coalitional game proposed by Abraham et al. (2006) the worth of a coalition ofvertices (neuronal structures) is defined by the number of strongly connected componentsin its induced subnetwork (within the whole brain network). We will illustrate thatthis coalitional game is counter-intuitive from a game theoretical point of view as itdoes not satisfy the two basic properties of superadditivity and monotonicity. Moreover,∗CentER and Departement of Econometrics and Operations Research Tilburg University, P.O. Box90153, 5000 LE Tilburg, The Netherlands†Correspondence to: m.musegaas@tilburguniversity.edu1the Shapley value of this game does not specify the relative influence of the neuronalstructures.In this paper we introduce an alternative coalitional game which resolves these issues.We discuss specific features of this game in small networks and apply our model to thelarge-scale brain networks considered by Kötter et al. (2007). Generally speaking, ourresults corroborate the findings of Kötter et al. (2007). Besides, since missing informa-tion on possible connections in a brain network is a common problem (cf. Kötter andStephan (2003)), we illustrate how our new approach allows for a direct incorporation ofprobabilistic considerations.2 Shapley ratings in brain networksA brain network is a directed graph (N,A) where N is a set of vertices, representing aset of neuronal structures, and A is a set of arcs, representing the connections betweenthe neuronal structures. Let A denote all ordered pairs (i, j) of vertices in N for whichthere exists a directed path from i to j in (N,A). A graph (N,A) is called stronglyconnected if for every two vertices i and j in N there is a directed path from i to j andfrom j to i in (N,A), i.e., if A contains all ordered pairs in N . The induced subgraph(S,A[S]) is a graph where a subset S ⊆ N is the set of vertices and A[S] is the set of arcsconsisting of any arc in A whose starting and end point are both in S. A strongly connectedcomponent is a maximal induced subgraph which is strongly connected, i.e., there is noother strongly connected subgraph containing this strongly connected component. LetSCC(N,A) denote the number of strongly connected components in graph (N,A).Example 2.1. Consider the brain network (N,A) with N = {1, 2, 3, 4} illustrated below.11234Note that (N,A) is strongly connected because for every vertex in the graph there existsa directed path to every other vertex. However, the subgraph induced by {1, 2, 3} is notstrongly connected and we haveA[{1, 2, 3}] = {(1, 2), (1, 3), (2, 1), (2, 3)}.Note that SCC({1, 2, 3}, A[{1, 2, 3}]) = 2 because the subgraph induced by {1, 2, 3} con-sists of two strongly connected components: the subgraphs induced by {1, 2} and {3}. 4A coalitional game is a pair (N, v) where N denotes a non-empty, finite set of playersand v is a function which assigns a number to each subset S ⊆ N (also called a coalition).By convention, v(∅) = 0. Abraham et al. (2006) introduced a coalitional game (N,wA)corresponding to a brain network (N,A) defined bywA(S) = SCC(S,A[S]),1This instance of a brain network is also used in Example 1 in Section 3.1 of Moretti (2013).2for all S ⊆ N . Hence, the worth of a coalition in wA is defined by the number of stronglyconnected components in its induced subgraph.Alternatively, we define the brain network game (N, vA) corresponding to (N,A) byvA(S) = |A[S]|,for all S ⊆ N . Hence, the worth of a coalition S in vA is defined by the number of orderedpairs (i, j) of vertices in S for which there exists a directed path from i to j in (S,A[S]).Two basic properties for coalitional games are superadditivity and monotonicity. Acoalitional game is called monotonic if the worth of a coalition increases when the coalitiongrows, and called superadditive if breaking up a coalition into parts does not pay. From agame theoretical perspective it is desirable that coalitional games satisfy these two basicproperties since they provide a clear incentive for cooperation in the grand coalition andthus provide a motivation to focus on fairly allocating the worth of the grand coalition.Unfortunately, these properties are not satisfied by the coalitional game (N,wA). Incontrast, the brain network game (N, vA) does satisfy monotonicity and superadditivity.This is illustrated in the following example.Example 2.2. Reconsider the brain network (N,A) presented in Example 2.1. The worthof every coalition in the games (N,wA) and (N, vA) is presented below.S {1} {2} {3} {4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2, 3, 4}wA(S) 1 1 1 1 1 2 2 2 2 2 2 2 3 1 1vA(S) 0 0 0 0 2 0 0 1 1 1 4 4 1 6 12Note that (N,wA) is not monotonic since {3, 4} ⊂ {2, 3, 4} but neverthelesswA({3, 4}) = 2 > 1 = wA({2, 3, 4}).Note that (N,wA) is also not superadditive since, e.g.,wA({1, 2}) + wA({3, 4}) = 3 > 1 = wA({1, 2, 3, 4}).It is readily checked that (N, vA) is both monotonic and superadditive. 4The Shapley value (cf. Shapley (1953)) of a coalitional game (N, v) is for all i ∈ Ndefined byΦi(v) =∑S⊆N\{i}pS (v(S ∪ {i})− v(S)),where pS =|S|!(|N |−|S|−1)!|N |! . Hence, the Shapley value looks at the marginal contributionsof a player to all possible coalitions. The weight pS is such that all marginal contribu-tions are weighted adequately to obtain an efficient allocation of the worth of the grandcoalition.In the context of coalitional games corresponding to brain networks, the Shapley valuecan be interpreted as a measure for the influence of a neuronal structure. Abraham et al.(2006) considered the Shapley value Φ(wA) as a rating for the neuronal structures in abrain network. Similarly, we consider the Shapley value Φ(vA) as a rating.3Example 2.3. Reconsider the coalitional games (N,wA) and (N, vA) of Example 2.2.The Shapley rating Φ(wA) is given byΦ(wA) =(12,−16, 13, 13), 2while the Shapley rating Φ(vA) is given byΦ(vA) =(216, 416, 256, 256),both determining a ranking (2, 3, 4, 1) or (2, 4, 3, 1) (there is a tie for the second highestranking). We note that a lower Shapley rating in wA indicates a higher influence in abrain network. On the contrary, a higher Shapley rating in vA indicates a higher influence.Since a Shapley rating in wA can be negative, as is the case in this example, it is notpossible to determine the relative influence of two vertices on the basis of Φ(wA). On theother hand, a Shapley rating in vA can not be negative by definition. Therefore, usingΦ(vA), we can say that the influence of vertex 2 in the brain network (N,A) is almosttwice as large as the influence of vertex 1. 4A common problem in the analysis of brain networks is the fact that it is not knownwhether some specific connections (arcs) are present or not (cf. Kötter and Stephan(2003)). Using a certain probabilistic knowledge about these unknown connections, thislack of information can readily be incorporated in the brain network game.We assume that each possible arc (i, j) is present with probability pij ∈ [0, 1]. Clearly,for each present arc we set pij = 1 and for each absent arc we set pij = 0. All proba-bilities are summarized into a vector p. Given such a vector p, we define the stochasticbrain network game (N, vp) in which the worth of a coalition equals the expected (in theprobabilistic sense) number of ordered pairs for which there exists a directed path in itsinduced subgraph.Example 2.4. Reconsider the brain network presented in Example 2.1. Only now sup-pose that the arcs (1, 4) and (3, 1) are present with probability p14 and p31, respectively.The complete corresponding vector p can be found below.(i, j) (1, 2) (1, 3) (1, 4) (2, 1) (2, 3) (2, 4) (3, 1) (3, 2) (3, 4) (4, 1) (4, 2) (4, 3)pij 1 0 p14 1 1 0 p31 0 1 0 1 0In total there are four possible brain networks. These different brain networks are illus-trated below and the corresponding probabilities for those networks are p14p31, (1−p14)p31,p14(1− p31) and (1− p14)(1− p31) for (a), (b), (c), and (d), respectively.1234(a) (N,A1)1234(b) (N,A2)1234(c) (N,A3)1234(d) (N,A4)2Because of a mistake in the worth of wA({1, 2, 3}), the Shapley value is incorrectly stated by Moretti(2013).4In order to calculate the worth of coalition {1, 3, 4} in the corresponding stochastic brainnetwork game (N, vp) we take the following weighted averagesvp({1, 3, 4}) = p14p31 · vA1({1, 3, 4}) + (1− p14)p31 · vA2({1, 3, 4})+ p14(1− p31) · vA3({1, 3, 4}) + (1− p14)(1− p31) · vA4({1, 3, 4})= p14p31 · 3 + (1− p14)p31 · 2 + p14(1− p31) · 2 + (1− p14)(1− p31) · 1= 1 + p14 + p31.The worth of every coalition is presented below.S {1} {2} {3} {4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1, 2, 3} {1, 2, 4} {1, 3, 4} {2, 3, 4} {1, 2, 3, 4}vp(S) 0 0 0 0 2 p31 p14 1 1 1 4 + 2p31 4 + 2p14 1 + p14 + p31 6 12The Shapley rating of the game (N, vp) is given byΦ1(vp) = 216+ 13p14 +13p31,Φ2(vp) = 416− 16p14 − 16p31,Φ3(vp) = 256− 12p14 +13p31,Φ4(vp) = 256+ 13p14 − 12p31.Fore example, if p14 =12and p31 =13, thenΦ(vp) =(21636, 4 136, 22536, 23036),with corresponding ranking (2, 4, 3, 1). 43 Results and discussionIn this section we apply the Shapley rating based on the brain network game (N, vA) tothe two large-scale brain networks considered by Kötter et al. (2007) and we compare theresults.The first large-scale brain network is the macaque visual cortex with thirty neuronalstructures as illustrated in Figure 1 of Kötter et al. (2007). The ranking of the fivebrain regions with the highest Shapley rating obtained by means of the coalitional games(N,wA) and (N, vA) can be found below in (a) and (b) respectively.(a) Top 5 of Φ(wA)Ranking Brain region1. V42. FEF3. 464. V25. Vp(b) Top 5 of Φ(vA)Ranking Brain region1. V42. FEF3. Vp4. V25. 465Note that both ratings agree on the top 5; only with respect to the positions 3 and 5 thereare some minor differences.The entire Shapley rating Φ(vA) of the macaque visual cortex can be found in Figure 1in the appendix. Correspondingly, we can roughly divide the brain regions in five classesbased on the relative difference with the brain region with the highest Shapley rating. Weconsider the following five classes based on the differences in terms of percentage: 0%–5%,5%–10%, 10%–15%, 15%–20%, 20% and higher. The first class consists of the single brainregion V4 with the highest Shapley rating. The second class consists of the brain regionsFEF to TF as ordered in Figure 1 that differ 5%–10% with V 4. The brain regions in thethird class are MSTd to V3, in the fourth class we have MSTI to PITd and in the fifthclass we have the single brain region VOT with a relative influence which is 23% lowerthan that of V4.The second large-scale brain network is the macaque prefrontal cortex with twelveneuronal structures (as illustrated in Figure 3(a) of Kötter et al. (2007)). In this casethere is a lack of information about the presence or absence of nine connections. To getsome insight, Kötter et al. (2007) considered two extreme cases. First, they assume thatconnections with unknown presence are absent. Second, they assume that those connec-tions are present. For both extreme cases the Shapley ratings are calculated separately.Our stochastic brain network game provides a way to incorporate lack of information intoone Shapley rating on the basis of probabilistic information. For simplicity, we assumethat each connection with unknown presence is absent with probability 12. Note that,in case more information would become available, more adequate probabilities can bereadily inserted. Having the complete vector p of arc probabilities, one readily computesthe corresponding stochastic brain network game (N, vp) and the corresponding Shapleyrating Φ(vp). The ranking based on the Shapley rating Φ(vp) can be found below.Ranking Brain region1. 92. 243. 124. 105. 466. 257. 118. 8B9. 1310. 8A11. 4512. 146ReferencesAbraham, M., R. Kötter, A. Krumnack, and E. Wanke (2006). A connectivity rating forvertices in networks. Fourth IFIP International Conference on Theoretical ComputerScience-TCS , 283–298.Kötter, R., A. Reid, A. Krumnack, E. Wanke, and O. Sporns (2007). Shapley ratings inbrain networks. Frontiers in Neuroinformatics , 1, Article 2.Kötter, R. and K. Stephan (2003). Network participation indices: characterizing compo-nent roles for information processing in neural networks. Neural Networks , 16, 1261–1275.Moretti, S. (2013). On some open problems arising from the application of coalitionalgames in medicine. International Game Theory Review , 15(3), 1340020: 1–18.Neumann, J. von and O. Morgenstern (1944). Theory of Games and Economic Behavior.Princeton University Press, Princeton.Shapley, L. (1953). A value for n-person games. Annals of Mathematics Studies , 28, 307–317.7AppendixFigure 1: Shapley rating of the macaque visual cortex.8

A Note on Shapley Ratings in Brain Networks

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A Note on Shapley Ratings in Brain Networks

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