Sticky Seeding in Discrete-Time Reversible-Threshold Networks

When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of \textit{Sticky Interventions}: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is $\Omega(\ln (OPT) )$-hard to approximate and that maximizing conversion subject to a budget is $(1-\frac{1}{e})$-hard to approximate. Optimization heuristics which rely on many objective function evaluations may still be practical, particularly in relatively-sparse networks: we prove that the long-term impact of a Seed Set can be evaluated in $O(|E|^2)$ operations. For a more descriptive model variant in which some neighbors may be more influential than others, we show that under integer edge weights from $\{0,1,2,...,k\}$ objective function evaluation requires only $O(k|E|^2)$ operations. These operation bounds are based on improvements we give for bounds on time-steps-to-convergence under discrete-time reversible-threshold updates in networks.Comment: 19 pages, 2 figure

Clustered Networks Protect Cooperation Against Catastrophic Collapse

Assuming a society of conditional cooperators (or moody conditional cooperators), this computational study proposes a new perspective on the structural advantage of social network clustering. Previous work focused on how clustered structure might encourage initial outbreaks of cooperation or defend against invasion by a few defectors. Instead, we explore the ability of a societal structure to retain cooperative norms in the face of widespread disturbances. Such disturbances may abstractly describe hardships like famine and economic recession, or the random spatial placement of a substantial numbers of pure defectors (or round-1 defectors) among a spatially-structured population of players in a laboratory game, etc. As links in tightly-clustered societies are reallocated to distant contacts, we observe that a society becomes increasingly susceptible to catastrophic cascades of defection: mutually-beneficial cooperative norms can be destroyed completely by modest shocks of defection. In contrast, networks with higher clustering coefficients can withstand larger shocks of defection before being forced to catastrophically-low levels of cooperation. We observe a remarkably-linear protective effect of clustering coefficient that becomes active above a critical level of clustering. Notably, both the critical level and the slope of this dependence is higher for decision-rule parameterizations that correspond to higher costs of cooperation. Our modeling framework provides a simple way to reinterpret the counter-intuitive and widely-cited human experiments of Suri and Watts (2011) while also affirming the classical intuition that network clustering and higher levels of cooperation should be positively associated

Combinatorial Consequences of Relatives of the Lusternik-Schnirelmann-Borsuk Theorem

Call a set of 2n + k elements Kneser colored when its n-subsets are put into classes such that disjoint n-subsets are in different classes. Kneser showed that k + 2 classes are sufficient to Kneser-color the n-subsets of a 2n + k element set. There are several proofs that this same number is necessary which rely on fixed-point theorems related to the Lusternik-Schnirelmann- Borsuk (LSB) theorem. By employing generalizations of these theorems we expand the proofs mentioned to obtain proofs of an original result we call the Subcoloring theorem. The Subcoloring theorem asserts the existence of a partition of a Kneser-colored set that halves its classes in a special way. We demonstrate both a topological proof and a combinatorial proof of this main result. We present an original corollary that extends the Subcoloring theorem by providing bounds on the size of the pieces of the asserted partition. Throughout, we formulate our results both in combinatorial and graph theoretic terminology

Measuring the value of accurate link prediction for network seeding

Merging two classic questions The influence-maximization literature seeks small sets of individuals whose structural placement in the social network can drive large cascades of behavior. Optimization efforts to find the best seed set often assume perfect knowledge of the network topology. Unfortunately, social network links are rarely known in an exact way. When do seeding strategies based on less-than-accurate link prediction provide valuable insight? Our contribution We introduce optimized-against-a-sample (OAS) performance to measure the value of optimizing seeding based on a noisy observation of a network. Our computational study investigates OAS under several threshold-spread models in synthetic and real-world networks. Our focus is on measuring the value of imprecise link information. The level of investment in link prediction that is strategic appears to depend closely on spread model: in some parameter ranges investments in improving link prediction can pay substantial premiums in cascade size. For other ranges, such investments would be wasted. Several trends were remarkably consistent across topologies

How Low Can You Go? New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes

In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The k-planar crossing number of a graph cr k ( G) is the number of crossings required when every edge of G must be drawn in one of k distinct planes. It was shown in [2] that cr 2 ( Q 8 ) ≤ 256 which we improve to cr 2 ( Q 8 ) ≤ 128. Our approach highlights the relationship between symmetric drawings and the study of k-planar crossing numbers. We conclude with several open questions concerning this relationship

The LSB Theorem Implies the KKM Lemma

No abstract provided in this article

Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals

We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gr\"obner bases and Nullstellensatz certificates for the coloring ideals of general graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page