Maximizing the Spread of Influence via Generalized Degree Discount

<div><p>It is a crucial and fundamental issue to identify a small subset of influential spreaders that can control the spreading process in networks. In previous studies, a degree-based heuristic called <i>DegreeDiscount</i> has been shown to effectively identify multiple influential spreaders and has severed as a benchmark method. However, the basic assumption of <i>DegreeDiscount</i> is not adequate, because it treats all the nodes equally without any differences. To consider a general situation in real world networks, a novel heuristic method named <i>GeneralizedDegreeDiscount</i> is proposed in this paper as an effective extension of original method. In our method, the status of a node is defined as a probability of not being influenced by any of its neighbors, and an index generalized discounted degree of one node is presented to measure the expected number of nodes it can influence. Then the spreaders are selected sequentially upon its generalized discounted degree in current network. Empirical experiments are conducted on four real networks, and the results show that the spreaders identified by our approach are more influential than several benchmark methods. Finally, we analyze the relationship between our method and three common degree-based methods.</p></div

The diagram of <i>DegreeDiscount</i>.

<p>Nodes filled by gray denote the selected spreaders, and others denote normal nodes.</p

The basic topological features of four real networks.

<p><i>N</i> and <i>M</i> are the numbers of nodes and edges. 〈<i>k</i>〉 is the average degree. <i>d</i>_<i>max</i> denotes the network diameter and 〈<i>d</i>〉 denotes the average shortest path length. <i>r</i> and <i>cc</i> are the assortative coefficient and clustering coefficient, respectively.</p

The spreading influence of GeneralizedDegreeDiscount and three adaptive centrality-based methods under different effective spreading rates.

<p>The numbers of spreaders are 100 in all networks, and the results are obtained by averaging over 200 implementations of the SIR model.</p

The spreading influence of nine methods on four networks under different effective spreading rates.

<p>The parameters are <i>λ</i> = 1.1, <i>q</i> = 1/〈<i>k</i>〉 for all networks, and all results are obtained by averaging over 200 implementations of the SIR model.</p

The similarities between <i>GeneralizedDegreeDiscount</i> and <i>Degree</i>, <i>SingleDiscount</i>, <i>DegreeDiscount</i>.

<p>The fraction of spreaders is 1%.</p

The goodness for fitting the in-degree distributions of some citation networks by the power-law function <i>f</i>(<i>k</i>) = <i>ak</i>⁻².

<p>The goodness for fitting the in-degree distributions of some citation networks by the power-law function <i>f</i>(<i>k</i>) = <i>ak</i>⁻².</p

Out-degree distributions of the citation networks in Table 1 and the fitting curves of the distributions.

<p>The fitting model is the mixture generalized Poisson distribution (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120687#pone.0120687.e032" target="_blank">Equation (14)</a>).</p

Empirical citation networks.

<p>Empirical citation networks.</p

The in- and out-degree distributions of a network generated by the CC model.

<p>The functions of the CC model are set as follows: <i>N</i>(<i>t</i>) = [e^0.1<i>t</i>], </p><p></p><p><mo stretchy="false">∣</mo></p><p><mi>D</mi><mi>i</mi></p><mo stretchy="false">∣</mo><mo>=</mo><p></p><p><mn>0</mn><mo>.</mo><mn>15</mn><mi>β</mi><mo stretchy="false">(</mo></p><p><mi>θ</mi><mi>i</mi></p><mo stretchy="false">)</mo><p></p><p><mo stretchy="false">[</mo></p><p>e</p><p><mn>0</mn><mo>.</mo><mn>1</mn></p><p><mi>t</mi><mi>i</mi></p><p></p><p></p><mo stretchy="false">]</mo><p></p><p></p><p></p><p></p>, and <i>β</i>(⋅) is given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120687#pone.0120687.e005" target="_blank">Equation (2)</a>. The fitting functions in Panel (a) are the Poisson distribution <p></p><p><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo></p><p><mi>a</mi><mi>k</mi></p><p></p><p>e</p><p><mo>−</mo><mi>a</mi></p><p></p><p><mi>k</mi><mo>!</mo></p><p></p><p></p><p></p> and the mixture Poisson distribution given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0120687#pone.0120687.e031" target="_blank">Equation (13)</a>. The fitting functions in Panel (b) are the power-law functions <i>f</i>(<i>k</i>) = <i>ak</i>⁻² and <p></p><p><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo></p><p></p><p></p><p><mi>k</mi></p><p><mo>−</mo><mi>γ</mi></p><p></p><p></p><p></p><p></p><p><mo>∑</mo></p><p><mi>n</mi><mo>=</mo><mn>0</mn></p><mi>∞</mi><p></p><p></p><p></p><p><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo></p><p><mi>x</mi></p><p><mi>min</mi></p><p></p><mo stretchy="false">)</mo><p></p><p><mo>−</mo><mi>γ</mi></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p>.<p></p