Computational aspects of optimal strategic network diffusion (original) (raw)

Computational Complexity of Competitive Diffusion on (Un) weighted Graphs

Consider an undirected graph modeling a social network, where the vertices represent users, and the edges do connections among them. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her opinion, and then it spreads along the edges in the graphs. The objective of every player is to maximize the number of vertices the opinion infects. In this paper, we investigate a computational problem of asking whether a pure Nash equilibrium exists in the competitive diffusion game on unweighed and weighted graphs, and present several negative and positive results. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forest of paths with arbitrary weights is solvable in pseudo-polynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem for chain, cochain, and threshold graphs with arbitrary integer weights is solvable in polynomial time.

Competitive Diffusion on Weighted Graphs

Lecture Notes in Computer Science, 2015

Consider an undirected and vertex-weighted graph modeling a social network, where the vertices represent individuals, the edges do connections among them, and weights do levels of importance of individuals. In the competitive diffusion game, each of a number of players chooses a vertex as a seed to propagate his/her idea which spreads along the edges in the graph. The objective of every player is to maximize the sum of weights of vertices infected by his/her idea. In this paper, we study a computational problem of asking whether a pure Nash equilibrium exists in a given graph, and present several negative and positive results with regard to graph classes. We first prove that the problem is W[1]-hard when parameterized by the number of players even for unweighted graphs. We also show that the problem is NP-hard even for series-parallel graphs with positive integer weights, and is NP-hard even for forests with arbitrary integer weights. Furthermore, we show that the problem for forests of paths with arbitrary weights is solvable in pseudopolynomial time; and it is solvable in quadratic time if a given graph is unweighted. We also prove that the problem is solvable in polynomial time for chain graphs, cochain graphs, and threshold graphs with arbitrary integer weights.

Complexity of equilibrium in competitive diffusion games on social networks

Automatica, 2016

In this paper, we consider the competitive diffusion game, and study the existence of its pure-strategy Nash equilibrium when defined over general undirected networks. We first determine the set of pure-strategy Nash equilibria for two special but wellknown classes of networks, namely the lattice and the hypercube. Characterizing the utility of the players in terms of graphical distances of their initial seed placements to other nodes in the network, we show that in general networks the decision process on the existence of pure-strategy Nash equilibrium is an NP-hard problem. Following this, we provide some necessary conditions for a given profile to be a Nash equilibrium. Furthermore, we study players' utilities in the competitive diffusion game over Erdos-Renyi random graphs and show that as the size of the network grows, the utilities of the players are highly concentrated around their expectation, and are bounded below by some threshold based on the parameters of the network. Finally, we obtain a lower bound for the maximum social welfare of the game with two players, and study sub-modularity of the players' utilities.

Complexity of equilibrium in diffusion games on social networks

2014 American Control Conference, 2014

In this paper, we consider the competitive diffusion game, and study the existence of its pure-strategy Nash equilibrium when defined over general undirected networks. We first determine the set of pure-strategy Nash equilibria for two special but well-known classes of networks, namely the lattice and the hypercube. Characterizing the utility of the players in terms of graphical distances of their initial seed placements to other nodes in the network, we show that in general networks the decision process on the existence of purestrategy Nash equilibrium is an NP-hard problem. Following this, we provide some necessary conditions for a given profile to be a Nash equilibrium. Furthermore, we study players' utilities in the competitive diffusion game over Erdos-Renyi random graphs and show that as the size of the network grows, the utilities of the players are highly concentrated around their expectation, and are bounded below by some threshold based on the parameters of the network. Finally, we obtain a lower bound for the maximum social welfare of the game with two players, and study sub-modularity of the players' utilities.

Optimal random walks in complex networks with limited information

2010

Maximization of the entropy rate is an important issue to design diffusion processes aiming at a well-mixed state. We demonstrate that it is possible to construct maximal-entropy random walks with only local information on the graph structure. In particular, we show that an almost maximal-entropy random walk is obtained when the step probabilities are proportional to a power of the degree of the target node, with an exponent α that depends on the degree-degree correlations and is equal to 1 in uncorrelated graphs.

Spread of influence in weighted networks under time and budget constraints

Theoretical Computer Science, 2015

Given a network represented by a weighted directed graph G, we consider the problem of finding a bounded cost set of nodes S such that the influence spreading from S in G, within a given time bound, is as large as possible. The dynamic that governs the spread of influence is the following: initially only elements in S are influenced; subsequently at each round, the set of influenced elements is augmented by all nodes in the network that have a sufficiently large number of already influenced neighbors. We prove that the problem is NP-hard, even in simple networks like complete graphs and trees. We also derive a series of positive results. We present exact pseudo-polynomial time algorithms for general trees, that become polynomial time in case the trees are unweighted. This last result improves on previously published results. We also design polynomial time algorithms for general weighted paths and cycles, and for unweighted complete graphs.

Maximizing Diffusion on Dynamic Social Networks

2009

The influence maximization problem is an important one in social network analysis, with applications from marketing to epidemiology. The task is to select some subset of the nodes in the network which, when activated, will spread the activation to the greatest portion of the rest of the network as quickly as possible. Since exact solutions are computationally intractable greedy approximation algorithms have been developed. However, such methods have only been tested on static social networks, or those in which the edges do not change while diffusion is occurring on the network. This is despite the fact that many social networks exhibit strongly dynamic behavior. Applying the heuristics used for static networks to dynamic ones is not straight forward, since the metrics typically used to judge the influence of nodes are not well defined when edges are changing. This paper examines the use of several potential dynamic measures for use with greedy approximation algorithms. Both linear threshold and independent cascade models of diffusion are used, and networks are formed using random, preferential attachment and proximity-based paradigms.

Influence diffusion in social networks under time window constraints

Theoretical Computer Science, 2015

We study a combinatorial model of the spread of influence in networks that generalizes existing schemata recently proposed in the literature. In our model, agents change behaviors/opinions on the basis of information collected from their neighbors in a time interval of bounded size whereas agents are assumed to have unbounded memory in previously studied scenarios. In our mathematical framework, one is given a network G = (V, E), an integer value t(v) for each node v ∈ V , and a time window size λ. The goal is to determine a small set of nodes (target set) that influences the whole graph. The spread of influence proceeds in rounds as follows: initially all nodes in the target set are influenced; subsequently, in each round, any uninfluenced node v becomes influenced if the number of its neighbors that have been influenced in the previous λ rounds is greater than or equal to t(v). We prove that the problem of finding a minimum cardinality target set that influences the whole network G is hard to approximate within a polylogarithmic factor. On the positive side, we design exact polynomial time algorithms for paths, rings, trees, and complete graphs.

Optimal contact process on complex networks

2008

Contact processes on complex networks are a recent subject of study in nonequilibrium statistical physics and they are also important to applied fields such as epidemiology and computer and communication networks. A basic issue concerns finding an optimal strategy for spreading.

Spread-It: A Strategic Game of Competitive Diffusion Through Social Networks

IEEE Transactions on Games, 2019

Diffusion of information is a key factor in many social and political situations. This work presents a strategic game called "Spread-It," which models the spread of information through social network structures. In the game, two competing parties must decide on the allocation and timing of their limited resources with the goal of increasing their influence in the network. Since present decisions affect the future level of network penetration, their effort allocation must be carefully planned. The work starts by defining the mathematical characteristics of the game, followed by analytical results derived by implementing several strategies for winning the game. Analyzing the experiments provides few observations regarding the role of influencers ('hubs') under different game conditions, the superiority of Monte Carlo tree search strategy over traditional game-tree search methods, and the budget required to guarantee the game's finalization.