A full version of this abstract can be found at
http://arxiv.org/abs/1301.0302.Cascading processes on a network have been studied in
a variety of disciplines, including computer science [3], biol-
ogy [4], sociology [2], and economics [5]. Much existing work
in this area is based on pre-existing models. However, recent
examinations of social networks { both analysis of large data
sets and experimental { have indicated that there may be
additional factors to consider that are not taken into account
by these models [1]..

Schroeder, Robert

Shakarian, Paulo

Simari, Gerardo I.

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveFaculty and Researcher Publications Faculty and Researcher Publications2013-05MANCaLog:  A Logic for Multi-AttributeNetwork Cascades (Extended Abstract)Shakarian, Paulohttp://hdl.handle.net/10945/38297MANCaLog: A Logic for Multi-Attribute Network Cascades(Extended Abstract)Paulo ShakarianNetwork Science Center andDept. of Electrical Engineeringand Computer ScienceU.S. Military AcademyWest Point, NY 10996paulo@shakarian.netGerardo I. SimariDept. of Computer ScienceUniversity of OxfordWolfson Building, Parks RoadOxford OX1 3QD, UKgerardo.simari@cs.ox.ac.ukRobert SchroederCORE LabDefense Analysis Dept.Naval Postgraduate SchoolMonterey, CA 93943rcschroe@nps.eduCategories and Subject DescriptorsI.2.4 [Artificial Intelligence]: Knowledge RepresentationFormalisms and Methods—Representation LanguagesGeneral TermsLanguages, AlgorithmsKeywordsComplex Networks, Cascades, Logic Programming1. INTRODUCTIONCascading processes on a network have been studied ina variety of disciplines, including computer science [3], biol-ogy [4], sociology [2], and economics [5]. Much existing workin this area is based on pre-existing models. However, recentexaminations of social networks – both analysis of large datasets and experimental – have indicated that there may beadditional factors to consider that are not taken into accountby these models [1]. In this paper we introduce MANCaLog,a logical framework designed to describe cascades in complexnetworks that meets seven desiderata we selected based on athorough review of the relevant literature. First, the frame-work must consider multiply labeled and weighted nodes andedges. This aspect is due to the fact that cascades in real-world networks do not only seem to depend on topologicalproperties (i.e., an individual adopts a behavior after a cer-tain number of his friends do) but also due to characteristicsof that individual as well. Second, time should be explicitlyrepresented, and (third) it should be non-Markovian, mean-ing that a node may choose to adopt or not adopt a behaviorbased on any previous time point (not simply the last one).Fourth, there must be some representation of uncertainty.Fifth, we must allow for competing cascades as has beenpreviously done in the classic work of [4]. Sixth, cascadesshould be able to be non-monotonic, meaning that the num-ber of nodes with a given property may increase or decreasein a given time period. Finally, such a framework should betractable and allow for the computation of the outcome ofAppears in: Proceedings of the 12th International Confer-ence on Autonomous Agents and Multiagent Systems (AA-MAS 2013), Ito, Jonker, Gini, and Shehory (eds.), May, 6–10, 2013,Saint Paul, Minnesota, USA.Copyright© 2013, International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.the cascade in polynomial time. In this extended abstract,we introduce the MANCaLog language, which meets all ofthese properties.12. FRAMEWORKWe assume that agents are arranged in a directed graph(or network)G = (V,E), where V is the set of nodes (agents)and E the set of edges (their relationships). We also assumea set of labels L, which is partitioned into two sets: fluentlabels Lf (labels that can change over time) and non-fluentlabels Lnf (labels that do not); labels can be applied to boththe nodes and edges of the network. We will use G = V ∪Eto denote the set of all components (nodes and edges) in thenetwork.Our logical language uses atoms, referring to labels andweights, to describe properties of the nodes and edges. Thefirst piece of the syntax is the network atom. Given la-bel L ∈ L and weight interval bnd ⊆ [0, 1], then 〈L, bnd〉is a network atom. An atom is fluent (resp., non-fluent)if L ∈ Lf (resp., L ∈ Lnf ). NA is the set of all possi-ble network atoms. The definition is intuitive: L representsa property of the vertex or edge, and associated with thisproperty is some weight that may have associated uncer-tainty – hence represented as an interval bnd , which can beopen or closed. An invalid bound is represented by ∅. A setof these atoms is a world. For a given world W , we im-pose the requirement that for each L ∈ L there is no morethan one atom of the form 〈L, bnd〉 in W (and bnd 6= ∅).A network formula over NA is defined using conjunction,disjunction, and negation in the usual way. If a formula con-tains only non-fluent (resp., fluent) atoms, it is a non-fluent(resp., fluent) formula. The satisfaction relationship is de-fined as follows. If f is an atom of the form 〈L, bnd〉 thenW |= f iff there exists 〈L, bnd ′〉 ∈ W s.t. bnd ′ ⊆ bnd . Thesatisfaction of conjunctions, disjunctions, and negations offormulas is then defined in the normal inductive manner.For some arbitrary label L ∈ L, we will use the nota-tion Tr = 〈L, [0, 1]〉 and F = 〈L, ∅〉 to represent a tautologyand contradiction, respectively. For ease of notation (andwithout loss of generality), we say that if there does notexist some bnd s.t. 〈L, bnd〉 ∈ W , then this implies that〈L, [0, 1]〉 ∈ W . The idea is to use MANCaLog to describehow properties (specified by labels) of the nodes in the net-work change over time. We assume that there is some nat-1A full version of this abstract can be found athttp://arxiv.org/abs/1301.0302.1175ural number tmax that specifies the total amount of time weare considering, and we use τ = {t | t ∈ [0, tmax ]} to denotethe set of all time points. How well a certain property canbe attributed to a node is based on a weight (to which thebnd bound in the network atom refers). As time progresses,a weight can either increase or decrease and/or become moreor less certain. Next, MANCaLog facts state that some net-work atom is true for a node or edge during certain times. If[t1, t2] ⊆ [0, tmax ], c ∈ G, and a ∈ NA, then (a, c) : [t1, t2] is aMANCaLog fact. A fact is fluent (resp., non-fluent) if atoma is fluent (resp., non-fluent). All non-fluent facts must beof the form (a, c) : [0, tmax ]. Let F be the set of all facts andFnf ,Ff be the set of all non-fluent and fluent facts, respec-tively. Likewise, we introduce integrity constraints (ICs) asfollows: given fluent network atom a and conjunction of net-work atoms b, an integrity constraint is of the form a ←↩ b.Intuitively, integrity constraint 〈L, bnd〉 ←↩ b means that ifat a certain time point a component (vertex or edge) of thenetwork has a set of properties specified by conjunction b,then at that same time the component’s weight for label Lmust be in interval bnd .We now define MANCaLog rules. The idea behind rules issimple: an agent that meets some criteria is influenced bythe set of its neighbors who possess certain properties. Theamount of influence exerted on an agent by its neighbors isspecified by an influence function, whose precise effects willbe described later on when we discuss the semantics. As aresult, a rule consists of four major parts: (i) an influencefunction, (ii) neighbor criteria, (iii) target criteria, and (iv) atarget. Intuitively, (i) specifies how the neighbors influencethe agent in question, (ii) specifies which of the neighborscan influence the agent, (iii) specifies the criteria that causethe agent to be influenced, and (iv) is the property of theagent that changes as a result of the influence. We will dis-cuss each of these parts in turn, and then define rules interms of these elements. First, an influence function, whichis a function ifl : N ×N → [0, 1] × [0, 1] that satisfies thefollowing two axioms: (1) it can be computed in PTIMEand (2) for x′ > x we have ifl(x′, y) ⊆ ifl(x, y). This func-tion takes the number of qualifying and eligible influencersand returns a bound on the new value for the weight of theproperty of the target node that changes. The next part ofa rule is the neighborhood criterion: (gedge, gnode, h)ifl .Formulas gnode and h are non-fluent/fluent formulas thatspecify the (non-fluent and fluent, respectively) criteria on agiven neighbor, while the non-fluent formula gedge specifiesthe non-fluent criteria on the directed edge from that neigh-bor to the node in question. The next component is the“target criteria”, which are the criteria that an agent mustsatisfy in order to be influenced by its neighbors. Ideas suchas “susceptibility” [1] can be integrated into our frameworkvia this component. We represent these criteria with a for-mula of non-fluent network atoms. The final component, the“target” (denoted with fluent label L), is simply the label ofthe target agent that is influenced by its neighbors. Alongwith ∆t, which specifies the time until the target is affected,we now have all the pieces to define a rule:r = L∆t← f, (gedge, gnode, h)iflNote that the target (also referred to as the head) of therule is a single label; essentially, the body of the rule charac-terizes a set of nodes, and this label is the one that is modi-fied for each node in this set. More specifically, the rule saysthat when certain conditions for an agent and its neighborsare met, the bnd bound for the network atom formed withlabel L on that agent changes. Later, in the semantics, weintroduce network interpretations, which map components(nodes and edges) of the network to worlds at a given pointin time. The rule dictates how this mapping changes in thenext time step. Hence a MANCaLog program, P , is a setof rules, facts, and integrity constraints s.t. each non-fluentfact F ∈ Fnf appears no more than once in the program. Pis the set of all programs.Our first semantic structure: the network interpreta-tion is a mapping of network components to sets of net-work atoms, NI : G → NA. We will use NI to denotethe set of all network interpretations. We note that not alllabels will necessarily apply to all nodes and edges in thenetwork. For instance, certain labels may describe a rela-tionship while others may only describe a property of anindividual in the network. If a given label L does not de-scribe a certain component c of the network, then in a validnetwork interpretation NI, 〈L, [0, 1]〉 ∈ NI(c). Now we candefine a MANCaLog interpretation as a mapping of naturalnumbers in the interval [0, tmax ] to network interpretations,i.e., I : N→ NI. Let I be the set of all possible interpreta-tions.In the full version of the paper, we formally define whatit means for an interpretation I to satisfy a program P .We then define the problems of consistency and entailment.Program P is consistent if there exists an interpretation thatsatisfies all elements in P . Likewise, P entails fact F iff forall models I of P , it holds that I |= F . In the full paper,we define an ordering over models and define the concept ofminimal model – the interpretation that assigns componentsof the network the tightest bound possible on the weight. Ifwe can find the minimal model, the questions of consistencyand entailment can be answered easily. We prove, using afixed-point operator (a technique common in logic program-ming) that the minimal model of a MANCaLog program canbe found in polynomial time. Currently, we are creating animplementation for this framework and designing methodsto automatically learn these programs from data.3. ACKNOWLEDGMENTSP.S. is supported by ARO project 2GDATXR042, andG.S. by (UK) EPSRC grant EP/J008346/1 – PrOQAW.4. REFERENCES[1] S. Aral and D. Walker. Identifying Influential andSusceptible Members of Social Networks. Science,337(6092):337–341, 2012.[2] M. Granovetter. Threshold models of collectivebehavior. The American Journal of Sociology,83(6):1420–1443, 1978.[3] D. Kempe, J. Kleinberg, and E. Tardos. Maximizingthe spread of influence through a social network. InProc. of KDD ’03, pages 137–146. ACM, 2003.[4] E. Lieberman, C. Hauert, and M. A. Nowak.Evolutionary dynamics on graphs. Nature,433(7023):312–316, 2005.[5] T. C. Schelling. Micromotives and Macrobehavior.W.W. Norton and Co., 1978.1176

MANCaLog:  A Logic for Multi-Attribute Network Cascades (Extended Abstract)

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MANCaLog: A Logic for Multi-Attribute Network Cascades (Extended Abstract)

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